Getting a UK passport renewed when you have dual nationality and a different name in your second country?

I have subsequently learned that the UK passport office will accept having two surnames in my Costa Rican passport as long as I provide a letter from the Costa Rican authorities such as an embassy that two surnames are required under Costa Rican law for citizenship and passports. They will then issue a UK passport with a note that I am also known as... I think this is a bit like having a maiden name included in a passport. It is a requirement to declare if you hold a non-British passport. I am just surprised that this is not a routine issue at the Passport Office. That fact that I have to go to the trouble of getting the Costa Rican authorities to inform the British authorities that Costa Ricans carry two surnames is astonishing. Thanks to all those who made helpful comments on this post; the problem will soon be solved

The official British guidance on documents required for a passport renewal from abroad says the following:

Uncancelled non-British passports: Please send us a colour photocopy of your non-British passport (every page including blank pages). We retain the right to ask for the original passport. We will let you know by email or post if we need you to send it to us.

Thus you could cancel your Costa Rican passport and then apply for a British passport without including your Costa Rican documents. Unless you've already declared your second nationality to the UK, they wouldn't even know you're a dual national and shouldn't have any further questions. Then as soon as you get your British passport you can renew your Costa Rican passport and live in peace for another 10 years.

Taking a straightforward reading of the question, I think you've got things back to front. What you have to do is request a new British passport with a change of name:

If you have dual citizenship (‘dual nationality’) and have a non-British passport, the name on your non-British passport must match the name and gender you want on your British passport.

(emphasis added).

So the key is to understand that what you want is a British passport with (from a British cultural perspective) your new double-barrelled surname.

If your heart is set on a renewed British passport with only one surname¹, I'm afraid that you made a mistake when you decided to acquire Costa Rican citizenship. As far as the British government is concerned, you have chosen to change your name.

¹ I don't take this as implicit in the question, but it seems that some people do.

According to the guidance published about changes of names on the UK government site, specifically the document "Annex A: use of names in passports", if an applicant is unable to meet the requirement of using one name for all purposes for legal reasons, an exception can be made:

6.5 There are individual categories of applicant who may experience restrictions on their ability to meet the identity requirements of one name for all official purposes. These are primarily people who have dual nationality and who hold, have held or can obtain in the future a passport issued by another country.

6.6 Subject to the applicant being able to satisfy the following requirements, a passport may be issued in the name requested even where it differs from the name on the passport issued by another country. The following categories may be given exceptional consideration:

i.The law in the applicant’s country of origin restricts or prevents a change of name. Where there is such a restriction, the applicant will be required to provide evidence from their country of origin that a change of name is not permissible;

[...]

6.7 In the case of points (i)and (iii) above, a (British) passport may be issued and an observation placed in the passport saying:

“The holder has a [country] passport, number [ ] issued on [date] in the name of [ ]. This passport is due to expire on [date].”

If Costa Rica genuinely will not let you drop the extra surname from your official name, and this limitation can be documented, it seems like that should be sufficient grounds for considering an exception when you apply for your passport. Costa Rica isn't strictly your "country of origin" but I suspect that should be read as your country of other nationality (perhaps the idea of a Brit wanting to get citizenship somewhere else is unthinkable to the government).